Processing math: 100%

Your data matches 19 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000217
Mp00083: Standard tableaux shapeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
St000217: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0]
=> [1] => 0
[[1,2]]
=> [2]
=> [1,0,1,0]
=> [1,2] => 0
[[1],[2]]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0
[[1,2,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[[1,3],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[[1,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[[1,2,3,4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[[1,3,4],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[[1,2,4],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[[1,2,3],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[[1,3],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[[1,2,3,4,5]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[[1,3,4,5],[2]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[[1,2,4,5],[3]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[[1,2,3,5],[4]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[[1,2,3,4],[5]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[[1,3,5],[2,4]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[[1,2,5],[3,4]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[[1,3,4],[2,5]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[[1,2,4],[3,5]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[[1,2,3],[4,5]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[[1,2,3,4,5,6]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
[[1,3,5,6],[2,4]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 1
Description
The number of occurrences of the pattern 312 in a permutation.
Matching statistic: St000437
Mp00083: Standard tableaux shapeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
St000437: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0]
=> [1] => ? = 0
[[1,2]]
=> [2]
=> [1,0,1,0]
=> [1,2] => 0
[[1],[2]]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0
[[1,2,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[[1,3],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[[1,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[[1,2,3,4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[[1,3,4],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[[1,2,4],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[[1,2,3],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[[1,3],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[[1,2,3,4,5]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[[1,3,4,5],[2]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[[1,2,4,5],[3]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[[1,2,3,5],[4]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[[1,2,3,4],[5]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[[1,3,5],[2,4]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[[1,2,5],[3,4]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[[1,3,4],[2,5]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[[1,2,4],[3,5]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[[1,2,3],[4,5]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[[1,2,3,4,5,6]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
[[1,3,5,6],[2,4]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 1
[[1,2,5,6],[3,4]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 1
Description
The number of occurrences of the pattern 312 or of the pattern 321 in a permutation.
Mp00084: Standard tableaux conjugateStandard tableaux
St000017: Standard tableaux ⟶ ℤResult quality: 67% values known / values provided: 91%distinct values known / distinct values provided: 67%
Values
[[1]]
=> [[1]]
=> 0
[[1,2]]
=> [[1],[2]]
=> 0
[[1],[2]]
=> [[1,2]]
=> 0
[[1,2,3]]
=> [[1],[2],[3]]
=> 0
[[1,3],[2]]
=> [[1,2],[3]]
=> 0
[[1,2],[3]]
=> [[1,3],[2]]
=> 0
[[1],[2],[3]]
=> [[1,2,3]]
=> 0
[[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 0
[[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 0
[[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> 0
[[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 0
[[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 1
[[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 1
[[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 0
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 0
[[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 0
[[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 0
[[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 0
[[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 0
[[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> 0
[[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> 0
[[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 0
[[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> 1
[[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 1
[[1,3,4],[2,5]]
=> [[1,2],[3,5],[4]]
=> 1
[[1,2,4],[3,5]]
=> [[1,3],[2,5],[4]]
=> 1
[[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 1
[[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 0
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> 0
[[1,2,5],[3],[4]]
=> [[1,3,4],[2],[5]]
=> 0
[[1,3,4],[2],[5]]
=> [[1,2,5],[3],[4]]
=> 0
[[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> 0
[[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 0
[[1,4],[2,5],[3]]
=> [[1,2,3],[4,5]]
=> 1
[[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
[[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> 1
[[1,3],[2,4],[5]]
=> [[1,2,5],[3,4]]
=> 1
[[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
[[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 0
[[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> 0
[[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> 0
[[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 0
[[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 0
[[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 0
[[1,2,4,5,6],[3]]
=> [[1,3],[2],[4],[5],[6]]
=> 0
[[1,2,3,5,6],[4]]
=> [[1,4],[2],[3],[5],[6]]
=> 0
[[1,2,3,4,6],[5]]
=> [[1,5],[2],[3],[4],[6]]
=> 0
[[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 0
[[1,3,5,6],[2,4]]
=> [[1,2],[3,4],[5],[6]]
=> 1
[[1,3,5,7,9],[2,4,6,8,10]]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> ? = 4
[[1,3,5,7,8],[2,4,6,9,10]]
=> [[1,2],[3,4],[5,6],[7,9],[8,10]]
=> ? = 4
[[1,3,5,6,9],[2,4,7,8,10]]
=> [[1,2],[3,4],[5,7],[6,8],[9,10]]
=> ? = 4
[[1,3,5,6,8],[2,4,7,9,10]]
=> [[1,2],[3,4],[5,7],[6,9],[8,10]]
=> ? = 4
[[1,3,5,6,7],[2,4,8,9,10]]
=> [[1,2],[3,4],[5,8],[6,9],[7,10]]
=> ? = 4
[[1,3,4,7,9],[2,5,6,8,10]]
=> [[1,2],[3,5],[4,6],[7,8],[9,10]]
=> ? = 4
[[1,3,4,7,8],[2,5,6,9,10]]
=> [[1,2],[3,5],[4,6],[7,9],[8,10]]
=> ? = 4
[[1,3,4,6,9],[2,5,7,8,10]]
=> [[1,2],[3,5],[4,7],[6,8],[9,10]]
=> ? = 4
[[1,3,4,6,8],[2,5,7,9,10]]
=> [[1,2],[3,5],[4,7],[6,9],[8,10]]
=> ? = 4
[[1,3,4,6,7],[2,5,8,9,10]]
=> [[1,2],[3,5],[4,8],[6,9],[7,10]]
=> ? = 4
[[1,3,4,5,9],[2,6,7,8,10]]
=> [[1,2],[3,6],[4,7],[5,8],[9,10]]
=> ? = 4
[[1,3,4,5,8],[2,6,7,9,10]]
=> [[1,2],[3,6],[4,7],[5,9],[8,10]]
=> ? = 4
[[1,3,4,5,7],[2,6,8,9,10]]
=> [[1,2],[3,6],[4,8],[5,9],[7,10]]
=> ? = 4
[[1,3,4,5,6],[2,7,8,9,10]]
=> [[1,2],[3,7],[4,8],[5,9],[6,10]]
=> ? = 4
[[1,2,5,7,9],[3,4,6,8,10]]
=> [[1,3],[2,4],[5,6],[7,8],[9,10]]
=> ? = 4
[[1,2,5,7,8],[3,4,6,9,10]]
=> [[1,3],[2,4],[5,6],[7,9],[8,10]]
=> ? = 4
[[1,2,5,6,9],[3,4,7,8,10]]
=> [[1,3],[2,4],[5,7],[6,8],[9,10]]
=> ? = 4
[[1,2,5,6,8],[3,4,7,9,10]]
=> [[1,3],[2,4],[5,7],[6,9],[8,10]]
=> ? = 4
[[1,2,5,6,7],[3,4,8,9,10]]
=> [[1,3],[2,4],[5,8],[6,9],[7,10]]
=> ? = 4
[[1,2,4,7,9],[3,5,6,8,10]]
=> [[1,3],[2,5],[4,6],[7,8],[9,10]]
=> ? = 4
[[1,2,4,7,8],[3,5,6,9,10]]
=> [[1,3],[2,5],[4,6],[7,9],[8,10]]
=> ? = 4
[[1,2,4,6,9],[3,5,7,8,10]]
=> [[1,3],[2,5],[4,7],[6,8],[9,10]]
=> ? = 4
[[1,2,4,6,8],[3,5,7,9,10]]
=> [[1,3],[2,5],[4,7],[6,9],[8,10]]
=> ? = 4
[[1,2,4,6,7],[3,5,8,9,10]]
=> [[1,3],[2,5],[4,8],[6,9],[7,10]]
=> ? = 4
[[1,2,4,5,9],[3,6,7,8,10]]
=> [[1,3],[2,6],[4,7],[5,8],[9,10]]
=> ? = 4
[[1,2,4,5,8],[3,6,7,9,10]]
=> [[1,3],[2,6],[4,7],[5,9],[8,10]]
=> ? = 4
[[1,2,4,5,7],[3,6,8,9,10]]
=> [[1,3],[2,6],[4,8],[5,9],[7,10]]
=> ? = 4
[[1,2,4,5,6],[3,7,8,9,10]]
=> [[1,3],[2,7],[4,8],[5,9],[6,10]]
=> ? = 4
[[1,2,3,7,9],[4,5,6,8,10]]
=> [[1,4],[2,5],[3,6],[7,8],[9,10]]
=> ? = 4
[[1,2,3,7,8],[4,5,6,9,10]]
=> [[1,4],[2,5],[3,6],[7,9],[8,10]]
=> ? = 4
[[1,2,3,6,9],[4,5,7,8,10]]
=> [[1,4],[2,5],[3,7],[6,8],[9,10]]
=> ? = 4
[[1,2,3,6,8],[4,5,7,9,10]]
=> [[1,4],[2,5],[3,7],[6,9],[8,10]]
=> ? = 4
[[1,2,3,6,7],[4,5,8,9,10]]
=> [[1,4],[2,5],[3,8],[6,9],[7,10]]
=> ? = 4
[[1,2,3,5,9],[4,6,7,8,10]]
=> [[1,4],[2,6],[3,7],[5,8],[9,10]]
=> ? = 4
[[1,2,3,5,8],[4,6,7,9,10]]
=> [[1,4],[2,6],[3,7],[5,9],[8,10]]
=> ? = 4
[[1,2,3,5,7],[4,6,8,9,10]]
=> [[1,4],[2,6],[3,8],[5,9],[7,10]]
=> ? = 4
[[1,2,3,5,6],[4,7,8,9,10]]
=> [[1,4],[2,7],[3,8],[5,9],[6,10]]
=> ? = 4
[[1,2,3,4,9],[5,6,7,8,10]]
=> [[1,5],[2,6],[3,7],[4,8],[9,10]]
=> ? = 4
[[1,2,3,4,8],[5,6,7,9,10]]
=> [[1,5],[2,6],[3,7],[4,9],[8,10]]
=> ? = 4
[[1,2,3,4,7],[5,6,8,9,10]]
=> [[1,5],[2,6],[3,8],[4,9],[7,10]]
=> ? = 4
[[1,2,3,4,6],[5,7,8,9,10]]
=> [[1,5],[2,7],[3,8],[4,9],[6,10]]
=> ? = 4
[[1,2,3,4,5],[6,7,8,9,10]]
=> [[1,6],[2,7],[3,8],[4,9],[5,10]]
=> ? = 4
[[1,2,3,4,5],[6,7,8,9]]
=> [[1,6],[2,7],[3,8],[4,9],[5]]
=> ? = 3
[[1,2,3,4],[5,6,7,8],[9]]
=> [[1,5,9],[2,6],[3,7],[4,8]]
=> ? = 3
[[1,2,3,4],[5,6,7],[8,9]]
=> [[1,5,8],[2,6,9],[3,7],[4]]
=> ? = 4
[[1,2,3],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9]]
=> ? = 6
[[1,2,3],[4,5,6],[7,8],[9]]
=> [[1,4,7,9],[2,5,8],[3,6]]
=> ? = 4
[[1,2,3],[4,5],[6,7],[8,9]]
=> [[1,4,6,8],[2,5,7,9],[3]]
=> ? = 6
[[1,2],[3,4],[5,6],[7,8],[9]]
=> [[1,3,5,7,9],[2,4,6,8]]
=> ? = 6
[[1,2,3,4],[5,6,7,8],[9,10]]
=> [[1,5,9],[2,6,10],[3,7],[4,8]]
=> ? = 5
Description
The number of inversions of a standard tableau. Let T be a tableau. An inversion is an attacking pair (c,d) of the shape of T (see [[St000016]] for a definition of this) such that the entry of c in T is greater than the entry of d.
Matching statistic: St000497
Mp00284: Standard tableaux rowsSet partitions
Mp00219: Set partitions inverse YipSet partitions
Mp00217: Set partitions Wachs-White-rho Set partitions
St000497: Set partitions ⟶ ℤResult quality: 40% values known / values provided: 40%distinct values known / distinct values provided: 44%
Values
[[1]]
=> {{1}}
=> {{1}}
=> {{1}}
=> ? = 0
[[1,2]]
=> {{1,2}}
=> {{1,2}}
=> {{1,2}}
=> 0
[[1],[2]]
=> {{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> 0
[[1,2,3]]
=> {{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 0
[[1,3],[2]]
=> {{1,3},{2}}
=> {{1},{2,3}}
=> {{1},{2,3}}
=> 0
[[1,2],[3]]
=> {{1,2},{3}}
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 0
[[1],[2],[3]]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[[1,2,3,4]]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[[1,3,4],[2]]
=> {{1,3,4},{2}}
=> {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 0
[[1,2,4],[3]]
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
[[1,2,3],[4]]
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 0
[[1,3],[2,4]]
=> {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 1
[[1,2],[3,4]]
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> {{1,2,4},{3}}
=> 1
[[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> 0
[[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 0
[[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 0
[[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> {{1},{2,3,4,5}}
=> {{1},{2,3,4,5}}
=> 0
[[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> {{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> 0
[[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> 0
[[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 0
[[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> {{1,4,5},{2,3}}
=> {{1,3},{2,4,5}}
=> 1
[[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 1
[[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> {{1,5},{2,3,4}}
=> {{1,3,4},{2,5}}
=> 1
[[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> 1
[[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> {{1,2,3,5},{4}}
=> 1
[[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> {{1},{2},{3,4,5}}
=> {{1},{2},{3,4,5}}
=> 0
[[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> {{1},{2,3},{4,5}}
=> 0
[[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 0
[[1,3,4],[2],[5]]
=> {{1,3,4},{2},{5}}
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 0
[[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> 0
[[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 0
[[1,4],[2,5],[3]]
=> {{1,4},{2,5},{3}}
=> {{1},{2,5},{3,4}}
=> {{1},{2,4},{3,5}}
=> 1
[[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> {{1},{2,3,5},{4}}
=> {{1},{2,3,5},{4}}
=> 1
[[1,2],[3,5],[4]]
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 1
[[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> {{1,3},{2,4},{5}}
=> 1
[[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> {{1,2,4},{3},{5}}
=> 1
[[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> {{1},{2},{3},{4,5}}
=> {{1},{2},{3},{4,5}}
=> 0
[[1,4],[2],[3],[5]]
=> {{1,4},{2},{3},{5}}
=> {{1},{2},{3,4},{5}}
=> {{1},{2},{3,4},{5}}
=> 0
[[1,3],[2],[4],[5]]
=> {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 0
[[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 0
[[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> 0
[[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> {{1},{2,3,4,5,6}}
=> {{1},{2,3,4,5,6}}
=> 0
[[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> {{1,2},{3,4,5,6}}
=> {{1,2},{3,4,5,6}}
=> 0
[[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> {{1,2,3},{4,5,6}}
=> {{1,2,3},{4,5,6}}
=> 0
[[1,2,3,4,6],[5]]
=> {{1,2,3,4,6},{5}}
=> {{1,2,3,4},{5,6}}
=> {{1,2,3,4},{5,6}}
=> 0
[[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> {{1,2,3,4,5},{6}}
=> {{1,2,3,4,5},{6}}
=> 0
[[1,3,5,6],[2,4]]
=> {{1,3,5,6},{2,4}}
=> {{1,4,5,6},{2,3}}
=> {{1,3},{2,4,5,6}}
=> 1
[[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> {{1,2,4,5,6},{3}}
=> {{1,2,4,5,6},{3}}
=> 1
[[1,3,5,7,8],[2,4,6]]
=> {{1,3,5,7,8},{2,4,6}}
=> {{1,4,5},{2,3,6,7,8}}
=> {{1,3,5},{2,4,6,7,8}}
=> ? = 2
[[1,2,5,7,8],[3,4,6]]
=> {{1,2,5,7,8},{3,4,6}}
=> {{1,2,4,5},{3,6,7,8}}
=> {{1,2,5},{3,4,6,7,8}}
=> ? = 2
[[1,3,5,6,8],[2,4,7]]
=> {{1,3,5,6,8},{2,4,7}}
=> {{1,4,5,6},{2,3,7,8}}
=> {{1,3,5,6},{2,4,7,8}}
=> ? = 2
[[1,2,5,6,8],[3,4,7]]
=> {{1,2,5,6,8},{3,4,7}}
=> {{1,2,4,5,6},{3,7,8}}
=> {{1,2,5,6},{3,4,7,8}}
=> ? = 2
[[1,3,4,6,8],[2,5,7]]
=> {{1,3,4,6,8},{2,5,7}}
=> {{1,5,6},{2,3,4,7,8}}
=> {{1,3,4,6},{2,5,7,8}}
=> ? = 2
[[1,2,4,6,8],[3,5,7]]
=> {{1,2,4,6,8},{3,5,7}}
=> {{1,2,5,6},{3,4,7,8}}
=> {{1,2,4,6},{3,5,7,8}}
=> ? = 2
[[1,2,3,6,8],[4,5,7]]
=> {{1,2,3,6,8},{4,5,7}}
=> {{1,2,3,5,6},{4,7,8}}
=> {{1,2,3,6},{4,5,7,8}}
=> ? = 2
[[1,3,5,6,7],[2,4,8]]
=> {{1,3,5,6,7},{2,4,8}}
=> {{1,4,5,6,7},{2,3,8}}
=> {{1,3,5,6,7},{2,4,8}}
=> ? = 2
[[1,2,5,6,7],[3,4,8]]
=> {{1,2,5,6,7},{3,4,8}}
=> {{1,2,4,5,6,7},{3,8}}
=> {{1,2,5,6,7},{3,4,8}}
=> ? = 2
[[1,3,4,6,7],[2,5,8]]
=> {{1,3,4,6,7},{2,5,8}}
=> {{1,5,6,7},{2,3,4,8}}
=> {{1,3,4,6,7},{2,5,8}}
=> ? = 2
[[1,2,4,6,7],[3,5,8]]
=> {{1,2,4,6,7},{3,5,8}}
=> {{1,2,5,6,7},{3,4,8}}
=> {{1,2,4,6,7},{3,5,8}}
=> ? = 2
[[1,2,3,6,7],[4,5,8]]
=> {{1,2,3,6,7},{4,5,8}}
=> {{1,2,3,5,6,7},{4,8}}
=> {{1,2,3,6,7},{4,5,8}}
=> ? = 2
[[1,3,4,5,7],[2,6,8]]
=> {{1,3,4,5,7},{2,6,8}}
=> {{1,6,7},{2,3,4,5,8}}
=> {{1,3,4,5,7},{2,6,8}}
=> ? = 2
[[1,2,4,5,7],[3,6,8]]
=> {{1,2,4,5,7},{3,6,8}}
=> {{1,2,6,7},{3,4,5,8}}
=> {{1,2,4,5,7},{3,6,8}}
=> ? = 2
[[1,2,3,5,7],[4,6,8]]
=> {{1,2,3,5,7},{4,6,8}}
=> {{1,2,3,6,7},{4,5,8}}
=> {{1,2,3,5,7},{4,6,8}}
=> ? = 2
[[1,2,3,4,7],[5,6,8]]
=> {{1,2,3,4,7},{5,6,8}}
=> {{1,2,3,4,6,7},{5,8}}
=> {{1,2,3,4,7},{5,6,8}}
=> ? = 2
[[1,3,4,5,6],[2,7,8]]
=> {{1,3,4,5,6},{2,7,8}}
=> ?
=> ?
=> ? = 2
[[1,2,4,5,6],[3,7,8]]
=> {{1,2,4,5,6},{3,7,8}}
=> {{1,2,7},{3,4,5,6,8}}
=> ?
=> ? = 2
[[1,3,5,7],[2,4,6,8]]
=> {{1,3,5,7},{2,4,6,8}}
=> {{1,4,5,8},{2,3,6,7}}
=> {{1,3,5,7},{2,4,6,8}}
=> ? = 3
[[1,2,5,7],[3,4,6,8]]
=> {{1,2,5,7},{3,4,6,8}}
=> {{1,2,4,5,8},{3,6,7}}
=> {{1,2,5,7},{3,4,6,8}}
=> ? = 3
[[1,3,4,7],[2,5,6,8]]
=> {{1,3,4,7},{2,5,6,8}}
=> {{1,5,8},{2,3,4,6,7}}
=> {{1,3,4,7},{2,5,6,8}}
=> ? = 3
[[1,2,4,7],[3,5,6,8]]
=> {{1,2,4,7},{3,5,6,8}}
=> {{1,2,5,8},{3,4,6,7}}
=> {{1,2,4,7},{3,5,6,8}}
=> ? = 3
[[1,2,3,7],[4,5,6,8]]
=> {{1,2,3,7},{4,5,6,8}}
=> {{1,2,3,5,8},{4,6,7}}
=> {{1,2,3,7},{4,5,6,8}}
=> ? = 3
[[1,3,5,6],[2,4,7,8]]
=> {{1,3,5,6},{2,4,7,8}}
=> {{1,4,5,6,8},{2,3,7}}
=> {{1,3,5,6,8},{2,4,7}}
=> ? = 3
[[1,2,5,6],[3,4,7,8]]
=> {{1,2,5,6},{3,4,7,8}}
=> {{1,2,4,5,6,8},{3,7}}
=> {{1,2,5,6,8},{3,4,7}}
=> ? = 3
[[1,3,4,6],[2,5,7,8]]
=> {{1,3,4,6},{2,5,7,8}}
=> {{1,5,6,8},{2,3,4,7}}
=> {{1,3,4,6,8},{2,5,7}}
=> ? = 3
[[1,2,4,6],[3,5,7,8]]
=> {{1,2,4,6},{3,5,7,8}}
=> {{1,2,5,6,8},{3,4,7}}
=> {{1,2,4,6,8},{3,5,7}}
=> ? = 3
[[1,2,3,6],[4,5,7,8]]
=> {{1,2,3,6},{4,5,7,8}}
=> {{1,2,3,5,6,8},{4,7}}
=> {{1,2,3,6,8},{4,5,7}}
=> ? = 3
[[1,3,4,5],[2,6,7,8]]
=> {{1,3,4,5},{2,6,7,8}}
=> {{1,6,8},{2,3,4,5,7}}
=> {{1,3,4,5,8},{2,6,7}}
=> ? = 3
[[1,2,4,5],[3,6,7,8]]
=> {{1,2,4,5},{3,6,7,8}}
=> {{1,2,6,8},{3,4,5,7}}
=> {{1,2,4,5,8},{3,6,7}}
=> ? = 3
[[1,2,3,5],[4,6,7,8]]
=> {{1,2,3,5},{4,6,7,8}}
=> {{1,2,3,6,8},{4,5,7}}
=> {{1,2,3,5,8},{4,6,7}}
=> ? = 3
[[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> {{1,2,3,4,6,8},{5,7}}
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 3
[[1,4,6,8],[2,5,7],[3]]
=> {{1,4,6,8},{2,5,7},{3}}
=> {{1},{2,5,6},{3,4,7,8}}
=> ?
=> ? = 2
[[1,3,6,8],[2,5,7],[4]]
=> {{1,3,6,8},{2,5,7},{4}}
=> {{1},{2,3,5,6},{4,7,8}}
=> {{1},{2,3,6},{4,5,7,8}}
=> ? = 2
[[1,2,6,8],[3,5,7],[4]]
=> {{1,2,6,8},{3,5,7},{4}}
=> {{1,2},{3,5,6},{4,7,8}}
=> {{1,2},{3,6},{4,5,7,8}}
=> ? = 2
[[1,3,6,8],[2,4,7],[5]]
=> {{1,3,6,8},{2,4,7},{5}}
=> {{1,4,7,8},{2,3},{5,6}}
=> {{1,3},{2,4,6},{5,7,8}}
=> ? = 2
[[1,2,6,8],[3,4,7],[5]]
=> {{1,2,6,8},{3,4,7},{5}}
=> {{1,2,4,7,8},{3},{5,6}}
=> {{1,2,4,6},{3},{5,7,8}}
=> ? = 2
[[1,4,5,8],[2,6,7],[3]]
=> {{1,4,5,8},{2,6,7},{3}}
=> {{1},{2,6},{3,4,5,7,8}}
=> {{1},{2,4,5,7,8},{3,6}}
=> ? = 2
[[1,3,5,8],[2,6,7],[4]]
=> {{1,3,5,8},{2,6,7},{4}}
=> ?
=> ?
=> ? = 2
[[1,2,5,8],[3,6,7],[4]]
=> {{1,2,5,8},{3,6,7},{4}}
=> {{1,2},{3,6},{4,5,7,8}}
=> {{1,2},{3,5,7,8},{4,6}}
=> ? = 2
[[1,3,4,8],[2,6,7],[5]]
=> {{1,3,4,8},{2,6,7},{5}}
=> {{1},{2,3,4,6},{5,7,8}}
=> {{1},{2,3,4,7,8},{5,6}}
=> ? = 2
[[1,2,4,8],[3,6,7],[5]]
=> {{1,2,4,8},{3,6,7},{5}}
=> ?
=> ?
=> ? = 2
[[1,2,3,8],[4,6,7],[5]]
=> {{1,2,3,8},{4,6,7},{5}}
=> {{1,2,3},{4,6},{5,7,8}}
=> {{1,2,3},{4,7,8},{5,6}}
=> ? = 2
[[1,3,5,8],[2,4,7],[6]]
=> {{1,3,5,8},{2,4,7},{6}}
=> {{1,4,5,7,8},{2,3},{6}}
=> {{1,3},{2,4,5,7,8},{6}}
=> ? = 2
[[1,2,5,8],[3,4,7],[6]]
=> {{1,2,5,8},{3,4,7},{6}}
=> {{1,2,4,5,7,8},{3},{6}}
=> {{1,2,4,5,7,8},{3},{6}}
=> ? = 2
[[1,3,4,8],[2,5,7],[6]]
=> {{1,3,4,8},{2,5,7},{6}}
=> {{1,5,7,8},{2,3,4},{6}}
=> ?
=> ? = 2
[[1,2,4,8],[3,5,7],[6]]
=> {{1,2,4,8},{3,5,7},{6}}
=> {{1,2,5,7,8},{3,4},{6}}
=> {{1,2,4},{3,5,7,8},{6}}
=> ? = 2
[[1,2,3,8],[4,5,7],[6]]
=> {{1,2,3,8},{4,5,7},{6}}
=> {{1,2,3,5,7,8},{4},{6}}
=> {{1,2,3,5,7,8},{4},{6}}
=> ? = 2
[[1,3,5,8],[2,4,6],[7]]
=> {{1,3,5,8},{2,4,6},{7}}
=> ?
=> ?
=> ? = 2
Description
The lcb statistic of a set partition. Let S=B1,,Bk be a set partition with ordered blocks Bi and with minBa<minBb for a<b. According to [1, Definition 3], a '''lcb''' (left-closer-bigger) of S is given by a pair i<j such that j=maxBb and iBa for a>b.
Mp00284: Standard tableaux rowsSet partitions
St000609: Set partitions ⟶ ℤResult quality: 39% values known / values provided: 39%distinct values known / distinct values provided: 44%
Values
[[1]]
=> {{1}}
=> ? = 0
[[1,2]]
=> {{1,2}}
=> 0
[[1],[2]]
=> {{1},{2}}
=> 0
[[1,2,3]]
=> {{1,2,3}}
=> 0
[[1,3],[2]]
=> {{1,3},{2}}
=> 0
[[1,2],[3]]
=> {{1,2},{3}}
=> 0
[[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 0
[[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 0
[[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[[1,3],[2,4]]
=> {{1,3},{2,4}}
=> 1
[[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1
[[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 0
[[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> 0
[[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0
[[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 0
[[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 0
[[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 0
[[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0
[[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> 1
[[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 1
[[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> 1
[[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> 1
[[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 0
[[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> 0
[[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> 0
[[1,3,4],[2],[5]]
=> {{1,3,4},{2},{5}}
=> 0
[[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> 0
[[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,4],[2,5],[3]]
=> {{1,4},{2,5},{3}}
=> 1
[[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 1
[[1,2],[3,5],[4]]
=> {{1,2},{3,5},{4}}
=> 1
[[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> 1
[[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 0
[[1,4],[2],[3],[5]]
=> {{1,4},{2},{3},{5}}
=> 0
[[1,3],[2],[4],[5]]
=> {{1,3},{2},{4},{5}}
=> 0
[[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> 0
[[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> 0
[[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> 0
[[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> 0
[[1,2,3,4,6],[5]]
=> {{1,2,3,4,6},{5}}
=> 0
[[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0
[[1,3,5,6],[2,4]]
=> {{1,3,5,6},{2,4}}
=> 1
[[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> 1
[[1,3,5,7,8],[2,4,6]]
=> {{1,3,5,7,8},{2,4,6}}
=> ? = 2
[[1,2,5,7,8],[3,4,6]]
=> {{1,2,5,7,8},{3,4,6}}
=> ? = 2
[[1,3,4,7,8],[2,5,6]]
=> {{1,3,4,7,8},{2,5,6}}
=> ? = 2
[[1,2,4,7,8],[3,5,6]]
=> {{1,2,4,7,8},{3,5,6}}
=> ? = 2
[[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[[1,3,5,6,8],[2,4,7]]
=> {{1,3,5,6,8},{2,4,7}}
=> ? = 2
[[1,2,5,6,8],[3,4,7]]
=> {{1,2,5,6,8},{3,4,7}}
=> ? = 2
[[1,3,4,6,8],[2,5,7]]
=> {{1,3,4,6,8},{2,5,7}}
=> ? = 2
[[1,2,4,6,8],[3,5,7]]
=> {{1,2,4,6,8},{3,5,7}}
=> ? = 2
[[1,2,3,6,8],[4,5,7]]
=> {{1,2,3,6,8},{4,5,7}}
=> ? = 2
[[1,3,4,5,8],[2,6,7]]
=> {{1,3,4,5,8},{2,6,7}}
=> ? = 2
[[1,2,4,5,8],[3,6,7]]
=> {{1,2,4,5,8},{3,6,7}}
=> ? = 2
[[1,2,3,5,8],[4,6,7]]
=> {{1,2,3,5,8},{4,6,7}}
=> ? = 2
[[1,2,3,4,8],[5,6,7]]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 2
[[1,3,5,6,7],[2,4,8]]
=> {{1,3,5,6,7},{2,4,8}}
=> ? = 2
[[1,2,5,6,7],[3,4,8]]
=> {{1,2,5,6,7},{3,4,8}}
=> ? = 2
[[1,3,4,6,7],[2,5,8]]
=> {{1,3,4,6,7},{2,5,8}}
=> ? = 2
[[1,2,4,6,7],[3,5,8]]
=> {{1,2,4,6,7},{3,5,8}}
=> ? = 2
[[1,2,3,6,7],[4,5,8]]
=> {{1,2,3,6,7},{4,5,8}}
=> ? = 2
[[1,3,4,5,7],[2,6,8]]
=> {{1,3,4,5,7},{2,6,8}}
=> ? = 2
[[1,2,4,5,7],[3,6,8]]
=> {{1,2,4,5,7},{3,6,8}}
=> ? = 2
[[1,2,3,5,7],[4,6,8]]
=> {{1,2,3,5,7},{4,6,8}}
=> ? = 2
[[1,2,3,4,7],[5,6,8]]
=> {{1,2,3,4,7},{5,6,8}}
=> ? = 2
[[1,3,4,5,6],[2,7,8]]
=> {{1,3,4,5,6},{2,7,8}}
=> ? = 2
[[1,2,4,5,6],[3,7,8]]
=> {{1,2,4,5,6},{3,7,8}}
=> ? = 2
[[1,2,3,5,6],[4,7,8]]
=> {{1,2,3,5,6},{4,7,8}}
=> ? = 2
[[1,2,3,4,6],[5,7,8]]
=> {{1,2,3,4,6},{5,7,8}}
=> ? = 2
[[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,5,7],[2,4,6,8]]
=> {{1,3,5,7},{2,4,6,8}}
=> ? = 3
[[1,2,5,7],[3,4,6,8]]
=> {{1,2,5,7},{3,4,6,8}}
=> ? = 3
[[1,3,4,7],[2,5,6,8]]
=> {{1,3,4,7},{2,5,6,8}}
=> ? = 3
[[1,2,4,7],[3,5,6,8]]
=> {{1,2,4,7},{3,5,6,8}}
=> ? = 3
[[1,2,3,7],[4,5,6,8]]
=> {{1,2,3,7},{4,5,6,8}}
=> ? = 3
[[1,3,5,6],[2,4,7,8]]
=> {{1,3,5,6},{2,4,7,8}}
=> ? = 3
[[1,2,5,6],[3,4,7,8]]
=> {{1,2,5,6},{3,4,7,8}}
=> ? = 3
[[1,3,4,6],[2,5,7,8]]
=> {{1,3,4,6},{2,5,7,8}}
=> ? = 3
[[1,2,4,6],[3,5,7,8]]
=> {{1,2,4,6},{3,5,7,8}}
=> ? = 3
[[1,2,3,6],[4,5,7,8]]
=> {{1,2,3,6},{4,5,7,8}}
=> ? = 3
[[1,3,4,5],[2,6,7,8]]
=> {{1,3,4,5},{2,6,7,8}}
=> ? = 3
[[1,2,4,5],[3,6,7,8]]
=> {{1,2,4,5},{3,6,7,8}}
=> ? = 3
[[1,2,3,5],[4,6,7,8]]
=> {{1,2,3,5},{4,6,7,8}}
=> ? = 3
[[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,4,6,8],[2,5,7],[3]]
=> {{1,4,6,8},{2,5,7},{3}}
=> ? = 2
[[1,3,6,8],[2,5,7],[4]]
=> {{1,3,6,8},{2,5,7},{4}}
=> ? = 2
[[1,2,6,8],[3,5,7],[4]]
=> {{1,2,6,8},{3,5,7},{4}}
=> ? = 2
[[1,3,6,8],[2,4,7],[5]]
=> {{1,3,6,8},{2,4,7},{5}}
=> ? = 2
[[1,2,6,8],[3,4,7],[5]]
=> {{1,2,6,8},{3,4,7},{5}}
=> ? = 2
[[1,4,5,8],[2,6,7],[3]]
=> {{1,4,5,8},{2,6,7},{3}}
=> ? = 2
[[1,3,5,8],[2,6,7],[4]]
=> {{1,3,5,8},{2,6,7},{4}}
=> ? = 2
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.
Matching statistic: St000491
Mp00284: Standard tableaux rowsSet partitions
Mp00219: Set partitions inverse YipSet partitions
Mp00215: Set partitions Wachs-WhiteSet partitions
St000491: Set partitions ⟶ ℤResult quality: 39% values known / values provided: 39%distinct values known / distinct values provided: 44%
Values
[[1]]
=> {{1}}
=> {{1}}
=> {{1}}
=> ? = 0
[[1,2]]
=> {{1,2}}
=> {{1,2}}
=> {{1,2}}
=> 0
[[1],[2]]
=> {{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> 0
[[1,2,3]]
=> {{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 0
[[1,3],[2]]
=> {{1,3},{2}}
=> {{1},{2,3}}
=> {{1,2},{3}}
=> 0
[[1,2],[3]]
=> {{1,2},{3}}
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 0
[[1],[2],[3]]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[[1,2,3,4]]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[[1,3,4],[2]]
=> {{1,3,4},{2}}
=> {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 0
[[1,2,4],[3]]
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
[[1,2,3],[4]]
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 0
[[1,3],[2,4]]
=> {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 1
[[1,2],[3,4]]
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> {{1,3},{2,4}}
=> 1
[[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 0
[[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 0
[[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 0
[[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> {{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> 0
[[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> {{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> 0
[[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 0
[[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 0
[[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> {{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> 1
[[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> {{1,2,4,5},{3}}
=> {{1,2,4},{3,5}}
=> 1
[[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> {{1,5},{2,3,4}}
=> {{1,5},{2,3,4}}
=> 1
[[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> {{1,4},{2,3,5}}
=> 1
[[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> {{1,3},{2,4,5}}
=> 1
[[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> {{1},{2},{3,4,5}}
=> {{1,2,3},{4},{5}}
=> 0
[[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> 0
[[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 0
[[1,3,4],[2],[5]]
=> {{1,3,4},{2},{5}}
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 0
[[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> 0
[[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 0
[[1,4],[2,5],[3]]
=> {{1,4},{2,5},{3}}
=> {{1},{2,5},{3,4}}
=> {{1,4},{2,3},{5}}
=> 1
[[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> {{1},{2,3,5},{4}}
=> {{1,3},{2,4},{5}}
=> 1
[[1,2],[3,5],[4]]
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> 1
[[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> {{1},{2,5},{3,4}}
=> 1
[[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> {{1},{2,4},{3,5}}
=> 1
[[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> {{1},{2},{3},{4,5}}
=> {{1,2},{3},{4},{5}}
=> 0
[[1,4],[2],[3],[5]]
=> {{1,4},{2},{3},{5}}
=> {{1},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5}}
=> 0
[[1,3],[2],[4],[5]]
=> {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> {{1},{2},{3,4},{5}}
=> 0
[[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> 0
[[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> 0
[[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> {{1},{2,3,4,5,6}}
=> {{1,2,3,4,5},{6}}
=> 0
[[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> {{1,2},{3,4,5,6}}
=> {{1,2,3,4},{5,6}}
=> 0
[[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> {{1,2,3},{4,5,6}}
=> {{1,2,3},{4,5,6}}
=> 0
[[1,2,3,4,6],[5]]
=> {{1,2,3,4,6},{5}}
=> {{1,2,3,4},{5,6}}
=> {{1,2},{3,4,5,6}}
=> 0
[[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> {{1,2,3,4,5},{6}}
=> {{1},{2,3,4,5,6}}
=> 0
[[1,3,5,6],[2,4]]
=> {{1,3,5,6},{2,4}}
=> {{1,4,5,6},{2,3}}
=> {{1,2,3,6},{4,5}}
=> 1
[[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> {{1,2,4,5,6},{3}}
=> {{1,2,3,5},{4,6}}
=> 1
[[1,3,5,7,8],[2,4,6]]
=> {{1,3,5,7,8},{2,4,6}}
=> {{1,4,5},{2,3,6,7,8}}
=> {{1,2,3,6,8},{4,5,7}}
=> ? = 2
[[1,2,5,7,8],[3,4,6]]
=> {{1,2,5,7,8},{3,4,6}}
=> {{1,2,4,5},{3,6,7,8}}
=> {{1,2,3,6,7},{4,5,8}}
=> ? = 2
[[1,3,4,7,8],[2,5,6]]
=> {{1,3,4,7,8},{2,5,6}}
=> {{1,5},{2,3,4,6,7,8}}
=> {{1,2,3,5,8},{4,6,7}}
=> ? = 2
[[1,2,4,7,8],[3,5,6]]
=> {{1,2,4,7,8},{3,5,6}}
=> {{1,2,5},{3,4,6,7,8}}
=> {{1,2,3,5,7},{4,6,8}}
=> ? = 2
[[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> {{1,2,3,5},{4,6,7,8}}
=> {{1,2,3,5,6},{4,7,8}}
=> ? = 2
[[1,3,5,6,8],[2,4,7]]
=> {{1,3,5,6,8},{2,4,7}}
=> {{1,4,5,6},{2,3,7,8}}
=> {{1,2,6,8},{3,4,5,7}}
=> ? = 2
[[1,2,5,6,8],[3,4,7]]
=> {{1,2,5,6,8},{3,4,7}}
=> {{1,2,4,5,6},{3,7,8}}
=> {{1,2,6,7},{3,4,5,8}}
=> ? = 2
[[1,3,4,6,8],[2,5,7]]
=> {{1,3,4,6,8},{2,5,7}}
=> {{1,5,6},{2,3,4,7,8}}
=> {{1,2,5,8},{3,4,6,7}}
=> ? = 2
[[1,2,4,6,8],[3,5,7]]
=> {{1,2,4,6,8},{3,5,7}}
=> {{1,2,5,6},{3,4,7,8}}
=> {{1,2,5,7},{3,4,6,8}}
=> ? = 2
[[1,2,3,6,8],[4,5,7]]
=> {{1,2,3,6,8},{4,5,7}}
=> {{1,2,3,5,6},{4,7,8}}
=> {{1,2,5,6},{3,4,7,8}}
=> ? = 2
[[1,3,4,5,8],[2,6,7]]
=> {{1,3,4,5,8},{2,6,7}}
=> {{1,6},{2,3,4,5,7,8}}
=> {{1,2,4,8},{3,5,6,7}}
=> ? = 2
[[1,2,4,5,8],[3,6,7]]
=> {{1,2,4,5,8},{3,6,7}}
=> {{1,2,6},{3,4,5,7,8}}
=> {{1,2,4,7},{3,5,6,8}}
=> ? = 2
[[1,2,3,5,8],[4,6,7]]
=> {{1,2,3,5,8},{4,6,7}}
=> {{1,2,3,6},{4,5,7,8}}
=> {{1,2,4,6},{3,5,7,8}}
=> ? = 2
[[1,2,3,4,8],[5,6,7]]
=> {{1,2,3,4,8},{5,6,7}}
=> {{1,2,3,4,6},{5,7,8}}
=> {{1,2,4,5},{3,6,7,8}}
=> ? = 2
[[1,3,5,6,7],[2,4,8]]
=> {{1,3,5,6,7},{2,4,8}}
=> {{1,4,5,6,7},{2,3,8}}
=> {{1,6,8},{2,3,4,5,7}}
=> ? = 2
[[1,2,5,6,7],[3,4,8]]
=> {{1,2,5,6,7},{3,4,8}}
=> {{1,2,4,5,6,7},{3,8}}
=> {{1,6,7},{2,3,4,5,8}}
=> ? = 2
[[1,3,4,6,7],[2,5,8]]
=> {{1,3,4,6,7},{2,5,8}}
=> {{1,5,6,7},{2,3,4,8}}
=> {{1,5,8},{2,3,4,6,7}}
=> ? = 2
[[1,2,4,6,7],[3,5,8]]
=> {{1,2,4,6,7},{3,5,8}}
=> {{1,2,5,6,7},{3,4,8}}
=> {{1,5,7},{2,3,4,6,8}}
=> ? = 2
[[1,2,3,6,7],[4,5,8]]
=> {{1,2,3,6,7},{4,5,8}}
=> {{1,2,3,5,6,7},{4,8}}
=> {{1,5,6},{2,3,4,7,8}}
=> ? = 2
[[1,3,4,5,7],[2,6,8]]
=> {{1,3,4,5,7},{2,6,8}}
=> {{1,6,7},{2,3,4,5,8}}
=> {{1,4,8},{2,3,5,6,7}}
=> ? = 2
[[1,2,4,5,7],[3,6,8]]
=> {{1,2,4,5,7},{3,6,8}}
=> {{1,2,6,7},{3,4,5,8}}
=> {{1,4,7},{2,3,5,6,8}}
=> ? = 2
[[1,2,3,5,7],[4,6,8]]
=> {{1,2,3,5,7},{4,6,8}}
=> {{1,2,3,6,7},{4,5,8}}
=> {{1,4,6},{2,3,5,7,8}}
=> ? = 2
[[1,2,3,4,7],[5,6,8]]
=> {{1,2,3,4,7},{5,6,8}}
=> {{1,2,3,4,6,7},{5,8}}
=> {{1,4,5},{2,3,6,7,8}}
=> ? = 2
[[1,3,4,5,6],[2,7,8]]
=> {{1,3,4,5,6},{2,7,8}}
=> ?
=> ?
=> ? = 2
[[1,2,4,5,6],[3,7,8]]
=> {{1,2,4,5,6},{3,7,8}}
=> {{1,2,7},{3,4,5,6,8}}
=> {{1,3,7},{2,4,5,6,8}}
=> ? = 2
[[1,2,3,5,6],[4,7,8]]
=> {{1,2,3,5,6},{4,7,8}}
=> {{1,2,3,7},{4,5,6,8}}
=> {{1,3,6},{2,4,5,7,8}}
=> ? = 2
[[1,2,3,4,6],[5,7,8]]
=> {{1,2,3,4,6},{5,7,8}}
=> {{1,2,3,4,7},{5,6,8}}
=> {{1,3,5},{2,4,6,7,8}}
=> ? = 2
[[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> {{1,2,3,4,5,7},{6,8}}
=> {{1,3,4},{2,5,6,7,8}}
=> ? = 2
[[1,3,5,7],[2,4,6,8]]
=> {{1,3,5,7},{2,4,6,8}}
=> {{1,4,5,8},{2,3,6,7}}
=> {{1,4,6,8},{2,3,5,7}}
=> ? = 3
[[1,2,5,7],[3,4,6,8]]
=> {{1,2,5,7},{3,4,6,8}}
=> {{1,2,4,5,8},{3,6,7}}
=> {{1,4,6,7},{2,3,5,8}}
=> ? = 3
[[1,3,4,7],[2,5,6,8]]
=> {{1,3,4,7},{2,5,6,8}}
=> {{1,5,8},{2,3,4,6,7}}
=> {{1,4,5,8},{2,3,6,7}}
=> ? = 3
[[1,2,4,7],[3,5,6,8]]
=> {{1,2,4,7},{3,5,6,8}}
=> {{1,2,5,8},{3,4,6,7}}
=> {{1,4,5,7},{2,3,6,8}}
=> ? = 3
[[1,2,3,7],[4,5,6,8]]
=> {{1,2,3,7},{4,5,6,8}}
=> {{1,2,3,5,8},{4,6,7}}
=> {{1,4,5,6},{2,3,7,8}}
=> ? = 3
[[1,3,5,6],[2,4,7,8]]
=> {{1,3,5,6},{2,4,7,8}}
=> {{1,4,5,6,8},{2,3,7}}
=> {{1,3,6,8},{2,4,5,7}}
=> ? = 3
[[1,2,5,6],[3,4,7,8]]
=> {{1,2,5,6},{3,4,7,8}}
=> {{1,2,4,5,6,8},{3,7}}
=> {{1,3,6,7},{2,4,5,8}}
=> ? = 3
[[1,3,4,6],[2,5,7,8]]
=> {{1,3,4,6},{2,5,7,8}}
=> {{1,5,6,8},{2,3,4,7}}
=> {{1,3,5,8},{2,4,6,7}}
=> ? = 3
[[1,2,4,6],[3,5,7,8]]
=> {{1,2,4,6},{3,5,7,8}}
=> {{1,2,5,6,8},{3,4,7}}
=> {{1,3,5,7},{2,4,6,8}}
=> ? = 3
[[1,2,3,6],[4,5,7,8]]
=> {{1,2,3,6},{4,5,7,8}}
=> {{1,2,3,5,6,8},{4,7}}
=> {{1,3,5,6},{2,4,7,8}}
=> ? = 3
[[1,3,4,5],[2,6,7,8]]
=> {{1,3,4,5},{2,6,7,8}}
=> {{1,6,8},{2,3,4,5,7}}
=> {{1,3,4,8},{2,5,6,7}}
=> ? = 3
[[1,2,4,5],[3,6,7,8]]
=> {{1,2,4,5},{3,6,7,8}}
=> {{1,2,6,8},{3,4,5,7}}
=> {{1,3,4,7},{2,5,6,8}}
=> ? = 3
[[1,2,3,5],[4,6,7,8]]
=> {{1,2,3,5},{4,6,7,8}}
=> {{1,2,3,6,8},{4,5,7}}
=> {{1,3,4,6},{2,5,7,8}}
=> ? = 3
[[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> {{1,2,3,4,6,8},{5,7}}
=> {{1,3,4,5},{2,6,7,8}}
=> ? = 3
[[1,4,6,8],[2,5,7],[3]]
=> {{1,4,6,8},{2,5,7},{3}}
=> {{1},{2,5,6},{3,4,7,8}}
=> ?
=> ? = 2
[[1,3,6,8],[2,5,7],[4]]
=> {{1,3,6,8},{2,5,7},{4}}
=> {{1},{2,3,5,6},{4,7,8}}
=> {{1,2,5,6},{3,4,7},{8}}
=> ? = 2
[[1,2,6,8],[3,5,7],[4]]
=> {{1,2,6,8},{3,5,7},{4}}
=> {{1,2},{3,5,6},{4,7,8}}
=> {{1,2,5,6},{3,4},{7,8}}
=> ? = 2
[[1,3,6,8],[2,4,7],[5]]
=> {{1,3,6,8},{2,4,7},{5}}
=> {{1,4,7,8},{2,3},{5,6}}
=> {{1,2,5},{3,4,8},{6,7}}
=> ? = 2
[[1,2,6,8],[3,4,7],[5]]
=> {{1,2,6,8},{3,4,7},{5}}
=> {{1,2,4,7,8},{3},{5,6}}
=> {{1,2,5},{3,4,7},{6,8}}
=> ? = 2
[[1,4,5,8],[2,6,7],[3]]
=> {{1,4,5,8},{2,6,7},{3}}
=> {{1},{2,6},{3,4,5,7,8}}
=> {{1,2,4,7},{3,5,6},{8}}
=> ? = 2
[[1,3,5,8],[2,6,7],[4]]
=> {{1,3,5,8},{2,6,7},{4}}
=> ?
=> ?
=> ? = 2
Description
The number of inversions of a set partition. Let S=B1,,Bk be a set partition with ordered blocks Bi and with minBa<minBb for a<b. According to [1], see also [2,3], an inversion of S is given by a pair i>j such that j=minBb and iBa for a<b. This statistic is called '''ros''' in [1, Definition 3] for "right, opener, smaller". This is also the number of occurrences of the pattern {{1, 3}, {2}} such that 1 and 2 are minimal elements of blocks.
Matching statistic: St000589
Mp00083: Standard tableaux shapeInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00284: Standard tableaux rowsSet partitions
St000589: Set partitions ⟶ ℤResult quality: 39% values known / values provided: 39%distinct values known / distinct values provided: 44%
Values
[[1]]
=> [1]
=> [[1]]
=> {{1}}
=> ? = 0
[[1,2]]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[[1],[2]]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[[1,2,3]]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[[1,3],[2]]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[[1,2],[3]]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[[1,2,3,4]]
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[[1,3,4],[2]]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[[1,2,4],[3]]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[[1,2,3],[4]]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[[1,3],[2,4]]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[[1,3],[2],[4]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[[1,2],[3],[4]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[[1,2,3,4,5]]
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0
[[1,3,4,5],[2]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0
[[1,2,4,5],[3]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0
[[1,2,3,5],[4]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0
[[1,2,3,4],[5]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0
[[1,3,5],[2,4]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[[1,2,5],[3,4]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[[1,3,4],[2,5]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[[1,2,4],[3,5]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[[1,2,3],[4,5]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[[1,2,3,4,5,6]]
=> [6]
=> [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> 0
[[1,3,4,5,6],[2]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0
[[1,2,4,5,6],[3]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0
[[1,2,3,5,6],[4]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0
[[1,2,3,4,6],[5]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0
[[1,2,3,4,5],[6]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0
[[1,3,5,6],[2,4]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 1
[[1,2,5,6],[3,4]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 1
[[1,3,5,7,8],[2,4,6]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,5,7,8],[3,4,6]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,7,8],[2,5,6]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,7,8],[3,5,6]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,7,8],[4,5,6]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,5,6,8],[2,4,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,5,6,8],[3,4,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,6,8],[2,5,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,6,8],[3,5,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,6,8],[4,5,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,5,8],[2,6,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,5,8],[3,6,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,5,8],[4,6,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,4,8],[5,6,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,5,6,7],[2,4,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,5,6,7],[3,4,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,6,7],[2,5,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,6,7],[3,5,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,6,7],[4,5,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,5,7],[2,6,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,5,7],[3,6,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,5,7],[4,6,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,4,7],[5,6,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,5,6],[2,7,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,5,6],[3,7,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,5,6],[4,7,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,4,6],[5,7,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,4,5],[6,7,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,5,7],[2,4,6,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,2,5,7],[3,4,6,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,3,4,7],[2,5,6,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,2,4,7],[3,5,6,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,2,3,7],[4,5,6,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,3,5,6],[2,4,7,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,2,5,6],[3,4,7,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,3,4,6],[2,5,7,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,2,4,6],[3,5,7,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,2,3,6],[4,5,7,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,3,4,5],[2,6,7,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,2,4,5],[3,6,7,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,2,3,5],[4,6,7,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,2,3,4],[5,6,7,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,4,6,8],[2,5,7],[3]]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 2
[[1,3,6,8],[2,5,7],[4]]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 2
[[1,2,6,8],[3,5,7],[4]]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 2
[[1,3,6,8],[2,4,7],[5]]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 2
[[1,2,6,8],[3,4,7],[5]]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 2
[[1,4,5,8],[2,6,7],[3]]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 2
[[1,3,5,8],[2,6,7],[4]]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 2
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block.
Matching statistic: St000612
Mp00083: Standard tableaux shapeInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00284: Standard tableaux rowsSet partitions
St000612: Set partitions ⟶ ℤResult quality: 39% values known / values provided: 39%distinct values known / distinct values provided: 44%
Values
[[1]]
=> [1]
=> [[1]]
=> {{1}}
=> ? = 0
[[1,2]]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[[1],[2]]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[[1,2,3]]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[[1,3],[2]]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[[1,2],[3]]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[[1,2,3,4]]
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[[1,3,4],[2]]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[[1,2,4],[3]]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[[1,2,3],[4]]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[[1,3],[2,4]]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[[1,3],[2],[4]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[[1,2],[3],[4]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[[1,2,3,4,5]]
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0
[[1,3,4,5],[2]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0
[[1,2,4,5],[3]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0
[[1,2,3,5],[4]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0
[[1,2,3,4],[5]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0
[[1,3,5],[2,4]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[[1,2,5],[3,4]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[[1,3,4],[2,5]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[[1,2,4],[3,5]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[[1,2,3],[4,5]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[[1,2,3,4,5,6]]
=> [6]
=> [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> 0
[[1,3,4,5,6],[2]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0
[[1,2,4,5,6],[3]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0
[[1,2,3,5,6],[4]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0
[[1,2,3,4,6],[5]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0
[[1,2,3,4,5],[6]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0
[[1,3,5,6],[2,4]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 1
[[1,2,5,6],[3,4]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 1
[[1,3,5,7,8],[2,4,6]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,5,7,8],[3,4,6]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,7,8],[2,5,6]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,7,8],[3,5,6]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,7,8],[4,5,6]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,5,6,8],[2,4,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,5,6,8],[3,4,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,6,8],[2,5,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,6,8],[3,5,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,6,8],[4,5,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,5,8],[2,6,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,5,8],[3,6,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,5,8],[4,6,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,4,8],[5,6,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,5,6,7],[2,4,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,5,6,7],[3,4,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,6,7],[2,5,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,6,7],[3,5,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,6,7],[4,5,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,5,7],[2,6,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,5,7],[3,6,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,5,7],[4,6,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,4,7],[5,6,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,5,6],[2,7,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,5,6],[3,7,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,5,6],[4,7,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,4,6],[5,7,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,4,5],[6,7,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,5,7],[2,4,6,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,2,5,7],[3,4,6,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,3,4,7],[2,5,6,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,2,4,7],[3,5,6,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,2,3,7],[4,5,6,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,3,5,6],[2,4,7,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,2,5,6],[3,4,7,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,3,4,6],[2,5,7,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,2,4,6],[3,5,7,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,2,3,6],[4,5,7,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,3,4,5],[2,6,7,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,2,4,5],[3,6,7,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,2,3,5],[4,6,7,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,2,3,4],[5,6,7,8]]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
[[1,4,6,8],[2,5,7],[3]]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 2
[[1,3,6,8],[2,5,7],[4]]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 2
[[1,2,6,8],[3,5,7],[4]]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 2
[[1,3,6,8],[2,4,7],[5]]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 2
[[1,2,6,8],[3,4,7],[5]]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 2
[[1,4,5,8],[2,6,7],[3]]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 2
[[1,3,5,8],[2,6,7],[4]]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 2
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block.
Mp00081: Standard tableaux reading word permutationPermutations
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00069: Permutations complementPermutations
St000516: Permutations ⟶ ℤResult quality: 20% values known / values provided: 20%distinct values known / distinct values provided: 44%
Values
[[1]]
=> [1] => [1] => [1] => ? = 0
[[1,2]]
=> [1,2] => [1,2] => [2,1] => 0
[[1],[2]]
=> [2,1] => [2,1] => [1,2] => 0
[[1,2,3]]
=> [1,2,3] => [1,3,2] => [3,1,2] => 0
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => [2,3,1] => 0
[[1,2],[3]]
=> [3,1,2] => [3,1,2] => [1,3,2] => 0
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [1,2,3] => 0
[[1,2,3,4]]
=> [1,2,3,4] => [1,4,3,2] => [4,1,2,3] => 0
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 0
[[1,2,4],[3]]
=> [3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 0
[[1,2,3],[4]]
=> [4,1,2,3] => [4,1,3,2] => [1,4,2,3] => 0
[[1,3],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 0
[[1,3],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 0
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,5,4,3] => [4,5,1,2,3] => 0
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,1,5,4,2] => [3,5,1,2,4] => 0
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [4,1,5,3,2] => [2,5,1,3,4] => 0
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [5,1,4,3,2] => [1,5,2,3,4] => 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,5,1,4,3] => [4,1,5,2,3] => 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,5,1,4,2] => [3,1,5,2,4] => 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,5,1,4,3] => [4,1,5,2,3] => 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,5,1,4,2] => [3,1,5,2,4] => 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [4,5,1,3,2] => [2,1,5,3,4] => 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,5,4] => [3,4,5,1,2] => 0
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [4,2,1,5,3] => [2,4,5,1,3] => 0
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [4,3,1,5,2] => [2,3,5,1,4] => 0
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [5,2,1,4,3] => [1,4,5,2,3] => 0
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [5,3,1,4,2] => [1,3,5,2,4] => 0
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [5,4,1,3,2] => [1,2,5,3,4] => 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,2,5,1,4] => [3,4,1,5,2] => 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [4,2,5,1,3] => [2,4,1,5,3] => 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,3,5,1,2] => [2,3,1,5,4] => 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [5,2,4,1,3] => [1,4,2,5,3] => 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [5,3,4,1,2] => [1,3,2,5,4] => 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => [2,3,4,5,1] => 0
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [5,3,2,1,4] => [1,3,4,5,2] => 0
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [5,4,2,1,3] => [1,2,4,5,3] => 0
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [1,2,3,5,4] => 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,6,5,4,3,2] => [6,1,2,3,4,5] => 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,6,5,4,3] => [5,6,1,2,3,4] => 0
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [3,1,6,5,4,2] => [4,6,1,2,3,5] => 0
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [4,1,6,5,3,2] => [3,6,1,2,4,5] => 0
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [5,1,6,4,3,2] => [2,6,1,3,4,5] => 0
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => [1,6,2,3,4,5] => 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,6,1,5,4,3] => [5,1,6,2,3,4] => 1
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [3,6,1,5,4,2] => [4,1,6,2,3,5] => 1
[[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 1
[[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [3,7,1,6,5,4,2] => [5,1,7,2,3,4,6] => ? = 1
[[1,3,4,6,7],[2,5]]
=> [2,5,1,3,4,6,7] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 1
[[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => [3,7,1,6,5,4,2] => [5,1,7,2,3,4,6] => ? = 1
[[1,2,3,6,7],[4,5]]
=> [4,5,1,2,3,6,7] => [4,7,1,6,5,3,2] => [4,1,7,2,3,5,6] => ? = 1
[[1,3,4,5,7],[2,6]]
=> [2,6,1,3,4,5,7] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 1
[[1,2,4,5,7],[3,6]]
=> [3,6,1,2,4,5,7] => [3,7,1,6,5,4,2] => [5,1,7,2,3,4,6] => ? = 1
[[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => [4,7,1,6,5,3,2] => [4,1,7,2,3,5,6] => ? = 1
[[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => [5,7,1,6,4,3,2] => [3,1,7,2,4,5,6] => ? = 1
[[1,3,4,5,6],[2,7]]
=> [2,7,1,3,4,5,6] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 1
[[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => [3,7,1,6,5,4,2] => [5,1,7,2,3,4,6] => ? = 1
[[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => [4,7,1,6,5,3,2] => [4,1,7,2,3,5,6] => ? = 1
[[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => [5,7,1,6,4,3,2] => [3,1,7,2,4,5,6] => ? = 1
[[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [6,7,1,5,4,3,2] => [2,1,7,3,4,5,6] => ? = 1
[[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => [2,7,6,1,5,4,3] => [6,1,2,7,3,4,5] => ? = 2
[[1,2,5,7],[3,4,6]]
=> [3,4,6,1,2,5,7] => [3,7,6,1,5,4,2] => [5,1,2,7,3,4,6] => ? = 2
[[1,3,4,7],[2,5,6]]
=> [2,5,6,1,3,4,7] => [2,7,6,1,5,4,3] => [6,1,2,7,3,4,5] => ? = 2
[[1,2,4,7],[3,5,6]]
=> [3,5,6,1,2,4,7] => [3,7,6,1,5,4,2] => [5,1,2,7,3,4,6] => ? = 2
[[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [4,7,6,1,5,3,2] => [4,1,2,7,3,5,6] => ? = 2
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [2,7,6,1,5,4,3] => [6,1,2,7,3,4,5] => ? = 2
[[1,2,5,6],[3,4,7]]
=> [3,4,7,1,2,5,6] => [3,7,6,1,5,4,2] => [5,1,2,7,3,4,6] => ? = 2
[[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [2,7,6,1,5,4,3] => [6,1,2,7,3,4,5] => ? = 2
[[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => [3,7,6,1,5,4,2] => [5,1,2,7,3,4,6] => ? = 2
[[1,2,3,6],[4,5,7]]
=> [4,5,7,1,2,3,6] => [4,7,6,1,5,3,2] => [4,1,2,7,3,5,6] => ? = 2
[[1,3,4,5],[2,6,7]]
=> [2,6,7,1,3,4,5] => [2,7,6,1,5,4,3] => [6,1,2,7,3,4,5] => ? = 2
[[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => [3,7,6,1,5,4,2] => [5,1,2,7,3,4,6] => ? = 2
[[1,2,3,5],[4,6,7]]
=> [4,6,7,1,2,3,5] => [4,7,6,1,5,3,2] => [4,1,2,7,3,5,6] => ? = 2
[[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [5,7,6,1,4,3,2] => [3,1,2,7,4,5,6] => ? = 2
[[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => [3,2,7,1,6,5,4] => [5,6,1,7,2,3,4] => ? = 1
[[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [4,2,7,1,6,5,3] => [4,6,1,7,2,3,5] => ? = 1
[[1,2,6,7],[3,5],[4]]
=> [4,3,5,1,2,6,7] => [4,3,7,1,6,5,2] => [4,5,1,7,2,3,6] => ? = 1
[[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => [5,2,7,1,6,4,3] => [3,6,1,7,2,4,5] => ? = 1
[[1,2,6,7],[3,4],[5]]
=> [5,3,4,1,2,6,7] => [5,3,7,1,6,4,2] => [3,5,1,7,2,4,6] => ? = 1
[[1,4,5,7],[2,6],[3]]
=> [3,2,6,1,4,5,7] => [3,2,7,1,6,5,4] => [5,6,1,7,2,3,4] => ? = 1
[[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => [4,2,7,1,6,5,3] => [4,6,1,7,2,3,5] => ? = 1
[[1,2,5,7],[3,6],[4]]
=> [4,3,6,1,2,5,7] => [4,3,7,1,6,5,2] => [4,5,1,7,2,3,6] => ? = 1
[[1,3,4,7],[2,6],[5]]
=> [5,2,6,1,3,4,7] => [5,2,7,1,6,4,3] => [3,6,1,7,2,4,5] => ? = 1
[[1,2,4,7],[3,6],[5]]
=> [5,3,6,1,2,4,7] => [5,3,7,1,6,4,2] => [3,5,1,7,2,4,6] => ? = 1
[[1,2,3,7],[4,6],[5]]
=> [5,4,6,1,2,3,7] => [5,4,7,1,6,3,2] => [3,4,1,7,2,5,6] => ? = 1
[[1,3,5,7],[2,4],[6]]
=> [6,2,4,1,3,5,7] => [6,2,7,1,5,4,3] => [2,6,1,7,3,4,5] => ? = 1
[[1,2,5,7],[3,4],[6]]
=> [6,3,4,1,2,5,7] => [6,3,7,1,5,4,2] => [2,5,1,7,3,4,6] => ? = 1
[[1,3,4,7],[2,5],[6]]
=> [6,2,5,1,3,4,7] => [6,2,7,1,5,4,3] => [2,6,1,7,3,4,5] => ? = 1
[[1,2,4,7],[3,5],[6]]
=> [6,3,5,1,2,4,7] => [6,3,7,1,5,4,2] => [2,5,1,7,3,4,6] => ? = 1
[[1,2,3,7],[4,5],[6]]
=> [6,4,5,1,2,3,7] => [6,4,7,1,5,3,2] => [2,4,1,7,3,5,6] => ? = 1
[[1,4,5,6],[2,7],[3]]
=> [3,2,7,1,4,5,6] => [3,2,7,1,6,5,4] => [5,6,1,7,2,3,4] => ? = 1
[[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => [4,2,7,1,6,5,3] => [4,6,1,7,2,3,5] => ? = 1
[[1,2,5,6],[3,7],[4]]
=> [4,3,7,1,2,5,6] => [4,3,7,1,6,5,2] => [4,5,1,7,2,3,6] => ? = 1
[[1,3,4,6],[2,7],[5]]
=> [5,2,7,1,3,4,6] => [5,2,7,1,6,4,3] => [3,6,1,7,2,4,5] => ? = 1
[[1,2,4,6],[3,7],[5]]
=> [5,3,7,1,2,4,6] => [5,3,7,1,6,4,2] => [3,5,1,7,2,4,6] => ? = 1
Description
The number of stretching pairs of a permutation. This is the number of pairs (i,j) with π(i)<i<j<π(j).
Matching statistic: St000355
Mp00081: Standard tableaux reading word permutationPermutations
Mp00061: Permutations to increasing treeBinary trees
Mp00014: Binary trees to 132-avoiding permutationPermutations
St000355: Permutations ⟶ ℤResult quality: 16% values known / values provided: 16%distinct values known / distinct values provided: 44%
Values
[[1]]
=> [1] => [.,.]
=> [1] => 0
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 0
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 0
[[1,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [3,1,2] => 0
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0
[[1,2,3,4]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[[1,3,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 0
[[1,2,4],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 0
[[1,2,3],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 0
[[1,3],[2,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0
[[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0
[[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 0
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 0
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 0
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [6,5,4,3,1,2] => 0
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [6,5,4,3,1,2] => 0
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [6,5,4,3,1,2] => 0
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [6,5,4,3,1,2] => 0
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [6,5,4,3,1,2] => 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [6,5,4,2,1,3] => 1
[[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,3,4,6,7],[2,5]]
=> [2,5,1,3,4,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,2,3,6,7],[4,5]]
=> [4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,3,4,5,7],[2,6]]
=> [2,6,1,3,4,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,2,4,5,7],[3,6]]
=> [3,6,1,2,4,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,3,4,5,6],[2,7]]
=> [2,7,1,3,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,2,5,7],[3,4,6]]
=> [3,4,6,1,2,5,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,3,4,7],[2,5,6]]
=> [2,5,6,1,3,4,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,2,4,7],[3,5,6]]
=> [3,5,6,1,2,4,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,2,5,6],[3,4,7]]
=> [3,4,7,1,2,5,6] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,2,3,6],[4,5,7]]
=> [4,5,7,1,2,3,6] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,3,4,5],[2,6,7]]
=> [2,6,7,1,3,4,5] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,2,3,5],[4,6,7]]
=> [4,6,7,1,2,3,5] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
[[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
[[1,2,6,7],[3,5],[4]]
=> [4,3,5,1,2,6,7] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
[[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
[[1,2,6,7],[3,4],[5]]
=> [5,3,4,1,2,6,7] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
[[1,4,5,7],[2,6],[3]]
=> [3,2,6,1,4,5,7] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
[[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
[[1,2,5,7],[3,6],[4]]
=> [4,3,6,1,2,5,7] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
[[1,3,4,7],[2,6],[5]]
=> [5,2,6,1,3,4,7] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
[[1,2,4,7],[3,6],[5]]
=> [5,3,6,1,2,4,7] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
[[1,2,3,7],[4,6],[5]]
=> [5,4,6,1,2,3,7] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
[[1,3,5,7],[2,4],[6]]
=> [6,2,4,1,3,5,7] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
[[1,2,5,7],[3,4],[6]]
=> [6,3,4,1,2,5,7] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
[[1,3,4,7],[2,5],[6]]
=> [6,2,5,1,3,4,7] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
[[1,2,4,7],[3,5],[6]]
=> [6,3,5,1,2,4,7] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
[[1,2,3,7],[4,5],[6]]
=> [6,4,5,1,2,3,7] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
[[1,4,5,6],[2,7],[3]]
=> [3,2,7,1,4,5,6] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
[[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
[[1,2,5,6],[3,7],[4]]
=> [4,3,7,1,2,5,6] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
[[1,3,4,6],[2,7],[5]]
=> [5,2,7,1,3,4,6] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
[[1,2,4,6],[3,7],[5]]
=> [5,3,7,1,2,4,6] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
[[1,2,3,6],[4,7],[5]]
=> [5,4,7,1,2,3,6] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1
Description
The number of occurrences of the pattern 21-3. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern 213.
The following 9 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001208The number of connected components of the quiver of A/T when T is the 1-tilting module corresponding to the permutation in the Auslander algebra A of K[x]/(xn). St001427The number of descents of a signed permutation. St001487The number of inner corners of a skew partition. St001960The number of descents of a permutation minus one if its first entry is not one. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation.